Subgroups — C₂×C₄²×D₂₂₃ #28384024

Order 14272.

498 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×D₂₂₃	C₂	1	3	yes	maximal
2	C₂×C₄×Dic₂₂₃	C₂	1	3	yes	maximal, complemented
3	C₄²×D₂₂₃	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₄²×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂³×D₂₂₃	C₂²	1	1	yes	characteristic
6	C₂²×Dic₂₂₃	C₂²	1	3	yes	—
7	C₂×C₄×D₂₂₃	C₄	1	24	yes	direct product factor
8	C₂×C₄×D₂₂₃	C₂²	1	12	yes	—
9	C₄×Dic₂₂₃	C₂²	1	12	yes	complemented
10	C₂²×C₄×C₂₂₃	C₂²	1	3	yes	—
11	C₄²×C₂₂₃	C₂²	1	4	yes	complemented
12	C₂²×D₂₂₃	C₂×C₄	1	12	yes	—
13	C₂²×D₂₂₃	C₂³	1	2	yes	—
14	C₂×Dic₂₂₃	C₂×C₄	1	12	yes	complemented
15	C₂×Dic₂₂₃	C₂³	1	6	yes	—
16	C₄×D₂₂₃	C₂×C₄	1	48	yes	direct product factor
17	C₂³×C₂₂₃	C₂³	1	1	yes	characteristic
18	C₂×C₄×C₂₂₃	C₂×C₄	1	12	yes	complemented
19	C₂×C₄×C₂₂₃	C₂³	1	6	yes	—
20	C₂²×C₂₂₃	C₂²×C₄	1	6	yes	—
21	C₂²×C₂₂₃	C₂⁴	1	1	yes	characteristic
22	C₂×D₂₂₃	C₄²	1	16	yes	direct product factor
23	C₂×D₂₂₃	C₂²×C₄	1	12	yes	—
24	C₄×C₂₂₃	C₂²×C₄	1	12	yes	complemented
25	Dic₂₂₃	C₂²×C₄	1	12	yes	complemented
26	C₂×C₂₂₃	C₂×C₄²	1	4	yes	complemented
27	C₂×C₂₂₃	C₂³×C₄	1	3	yes	—
28	D₂₂₃	C₂×C₄²	1	8	yes	direct product factor
29	C₂₂₃	C₂²×C₄²	1	1	yes	characteristic, complemented, Sylow
30	C₂²×C₄²	—	223	1	no	maximal, Sylow
31	C₂³×C₄	—	223	3	no	—
32	C₂×C₄²	—	223	3	no	—
33	C₂×C₄²	—	223	8	no	—
34	C₂×C₄²	D₂₂₃	1	1	yes	characteristic, direct product factor
35	C₂⁴	—	223	1	no	—
36	C₂²×C₄	—	223	3	no	—
37	C₂²×C₄	—	223	24	no	—
38	C₂²×C₄	—	223	12	no	—
39	C₂²×C₄	C₂×D₂₂₃	1	3	yes	—
40	C₄²	—	223	12	no	—
41	C₄²	C₂×D₂₂₃	1	4	yes	direct product factor
42	C₂³	—	223	12	no	—
43	C₂³	—	223	2	no	—
44	C₂³	C₂²×D₂₂₃	1	1	yes	characteristic
45	C₂×C₄	—	223	12	no	—
46	C₂×C₄	—	223	6	no	—
47	C₂×C₄	—	223	48	no	—
48	C₂×C₄	C₄×D₂₂₃	1	12	yes	direct product factor
49	C₂×C₄	C₂²×D₂₂₃	1	6	yes	—
50	C₂²	—	223	16	no	—
51	C₂²	—	223	12	no	—
52	C₂²	C₂×C₄×D₂₂₃	1	6	yes	—
53	C₂²	C₂³×D₂₂₃	1	1	yes	characteristic
54	C₄	—	223	12	no	—
55	C₄	C₂×C₄×D₂₂₃	1	12	yes	direct product factor
56	C₂	—	223	8	no	—
57	C₂	C₄²×D₂₂₃	1	4	yes	direct product factor
58	C₂	C₂²×C₄×D₂₂₃	1	3	yes	—

	subgroup	G/H
center, hypercenter	C₂×C₄²	D₂₂₃
commutator, hypocenter	C₂₂₃	C₂²×C₄²
Frattini	C₂²	C₂³×D₂₂₃
Fitting	C₂×C₄²×C₂₂₃	C₂
socle	C₂³×C₂₂₃	C₂³
prime radical	C₂³×D₂₂₃	C₂²
cosocle	C₂²×C₂₂₃	C₂⁴
LCS	G ▷ C₂₂₃
UCS	C₁ ◁ C₂×C₄²

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₂×C₄×Dic₂₂₃	C₂	semidirect
3. C₄²×D₂₂₃	C₂	direct (N × H)
4. C₂×C₄²×C₂₂₃	C₂	semidirect
7. C₂×C₄×D₂₂₃	C₄	direct (N × H)
9. C₄×Dic₂₂₃	C₂²	semidirect
11. C₄²×C₂₂₃	C₂²	semidirect
14. C₂×Dic₂₂₃	C₂×C₄	semidirect
16. C₄×D₂₂₃	C₂×C₄	direct (N × H)
18. C₂×C₄×C₂₂₃	C₂×C₄	semidirect
22. C₂×D₂₂₃	C₄²	direct (N × H)
24. C₄×C₂₂₃	C₂²×C₄	semidirect
25. Dic₂₂₃	C₂²×C₄	semidirect
26. C₂×C₂₂₃	C₂×C₄²	semidirect
28. D₂₂₃	C₂×C₄²	direct (N × H)
29. C₂₂₃	C₂²×C₄²	semidirect
34. C₂×C₄²	D₂₂₃	direct (N × H)
41. C₄²	C₂×D₂₂₃	direct (N × H)
48. C₂×C₄	C₄×D₂₂₃	direct (N × H)
55. C₄	C₂×C₄×D₂₂₃	direct (N × H)
57. C₂	C₄²×D₂₂₃	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₄×D₂₂₃	2
C₂×C₄×Dic₂₂₃	2
C₄²×D₂₂₃	2
C₂×C₄²×C₂₂₃	2
C₂²×C₄²	223

Subgroups — C2×C42×D223 #28384024

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄²×D₂₂₃ #28384024